We describe an explicit, simple, construction of large subsets of F^n, where F is a finite field, that have small intersection with every k-dimensional affine subspace. Interest in the explicit construction of such sets, termed 'subspace-evasive' sets, started in the work of Pudlak and Rodl (2004) who showed how such constructions over the binary field can be used to construct explicit Ramsey graphs. More recently, Guruswami (2011) showed that, over large finite fields (of size polynomial in n), subspace evasive sets can be used to obtain explicit list-decodable codes with optimal rate and constant list-size. We construct subspace evasive sets over large fields (polynomial in n) and use them to derive the applications to list-decodable codes discovered by Guruswami. Our construction employs both combinatoric and algebraic methods. (Joint work with S. Lovett)